Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs

TL;DR

This paper develops methods to compute connection probabilities for random spanning trees on surface-embedded graphs and applies these to determine the likelihood of loop-erased random walks passing through specific vertices in various lattices, confirming longstanding conjectures.

Contribution

It introduces a novel approach to calculate connection probabilities for spanning trees on surface-embedded graphs and applies it to analyze loop-erased random walks on different lattices.

Findings

Probability that LERW passes through (1,0) in Z^2 is 5/16

Confirmed conjecture about stationary sandpile density on Z^2

Computed LERW passage probabilities for triangular, honeycomb, and Z x R lattices

Abstract

We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the "intensity" of the loop-erased random walk in ${\mathbb Z}^2$, that is, the probability that the walk from (0,0) to infinity passes through a given vertex or edge. For example, the probability that it passes through (1,0) is 5/16; this confirms a conjecture from 1994 about the stationary sandpile density on ${\mathbb Z}^2$. We do the analogous computation for the triangular lattice, honeycomb lattice and ${\mathbb Z} \times {\mathbb R}$, for which the probabilities are 5/18, 13/36, and $1/4-1/\pi^2$ respectively.